A newsgroup is interested in constructing a 90% confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of the 530 randomly selected Americans surveyed, 390 were in favor of the initiative. Round answers to 4 decimal places where possible.

a.  With 90% confidence the proportion of all Americans who favor the new Green initiative is between 
 and 
.

b.  If many groups of 530 randomly selected Americans were surveyed, then a different confidence interval would be produced from each group. About 
 percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about 
 percent will not contain the true population proportion.

To calculate the 90% confidence interval for the proportion of Americans who favor the new Green initiative, we will use the formula for a confidence interval for a population proportion:

\[
\hat{p} \pm Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \(\hat{p}\) is the sample proportion,
- \(Z_{\alpha/2}\) is the critical value for the desired confidence level,
- \(n\) is the sample size.

### Step 1: Calculate the sample proportion (\(\hat{p}\))

\[
\hat{p} = \frac{390}{530} = 0.7358
\]

### Step 2: Find the critical value for a 90% confidence level
For a 90% confidence level, the critical value \(Z_{\alpha/2}\) is approximately 1.645.

### Step 3: Calculate the standard error
The standard error (SE) is given by:

\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.7358 \times (1 - 0.7358)}{530}} = \sqrt{\frac{0.7358 \times 0.2642}{530}} = \sqrt{\frac{0.1943}{530}} = \sqrt{0.0003667} = 0.0191
\]

### Step 4: Calculate the margin of error
The margin of error (ME) is:

\[
ME = Z_{\alpha/2} \times SE = 1.645 \times 0.0191 = 0.0314
\]

### Step 5: Construct the confidence interval
The 90% confidence interval is:

\[
\hat{p} \pm ME = 0.7358 \pm 0.0314
\]

So, the interval is:

\[
0.7358 - 0.0314 = 0.7044
\]
\[
0.7358 + 0.0314 = 0.7672
\]

Thus, the 90% confidence interval for the proportion of Americans who favor the new Green initiative is **between 0.7044 and 0.7672**.

### Answer to part (a):
\[
\boxed{0.7044 \text{ and } 0.7672}
\]

### Answer to part (b):
In a 90% confidence interval, we expect that 90% of the intervals will contain the true population proportion, while 10% will not. Therefore:

\[
\boxed{90} \text{ percent will contain the true proportion and } \boxed{10} \text{ percent will not.}
\]

---

Would you like more details on any part of the calculations? Here are five related questions you might consider:

1. How do you find the critical value for different confidence levels?
2. What is the interpretation of a confidence interval in real-world terms?
3. How does sample size affect the width of a confidence interval?
4. Can confidence intervals be calculated for different population parameters, like mean?
5. What are some common mistakes when interpreting confidence intervals?

**Tip**: The width of a confidence interval depends on both the confidence level and the sample size. Increasing the sample size reduces the width of the interval.

A newsgroup is interested in constructing a 90% confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of the 530 randomly selected Americans surveyed, 390 were in favor of the initiative. Round answers to 4 decimal places where possible.

a. With 90% confidence the proportion of all Americans who favor the new Green initiative is between _ and _.

b. If many groups of 530 randomly selected Americans were surveyed, then a different confidence interval would be produced from each group. About _ percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about _ percent will not contain the true population proportion.

Calculate a 90% confidence interval for the proportion of Americans in favor of a new Green initiative based on a sample of 530 individuals, with 390 expressing support. This problem covers key statistical concepts like population proportion, confidence intervals, and margin of error.

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